Let X = Y + Z where Y has density function f0(y) and Z is independent of Y with moments η

i

, η

ij

, η

ijk

,…. Show formally, that the density of X is given by fX (x) = EZ {f0 (x − Z)}.

Hence derive the series (5.2) by Taylor expansion of f0(x)• By taking η

i = 0, η

i,j

= 0, and f0(x) = ϕ(x; κ)derive the usual Edgeworth expansion for the density of X, taking care to group terms in the appropriate manner. (Davis, 1976).